Difference between with replacement and without replacement in probability
Answers
Answer:
Step-by-step explanation:
Sampling with replacement:
Consider a population of potato sacks, each of which has either 12, 13, 14, 15, 16, 17, or 18 potatoes, and all the values are equally likely. Suppose that, in this population, there is exactly one sack with each number. So the whole population has seven sacks. If I sample two with replacement, then I first pick one (say 14). I had a 1/7 probability of choosing that one. Then I replace it. Then I pick another. Every one of them still has 1/7 probability of being chosen. And there are exactly 49 different possibilities here (assuming we distinguish between the first and second.) They are: (12,12), (12,13), (12, 14), (12,15), (12,16), (12,17), (12,18), (13,12), (13,13), (13,14), etc.
Sampling without replacement:
Consider the same population of potato sacks, each of which has either 12, 13, 14, 15, 16, 17, or 18 potatoes, and all the values are equally likely. Suppose that, in this population, there is exactly one sack with each number. So the whole population has seven sacks. If I sample two without replacement, then I first pick one (say 14). I had a 1/7 probability of choosing that one. Then I pick another. At this point, there are only six possibilities: 12, 13, 15, 16, 17, and 18. So there are only 42 different possibilities here (again assuming that we distinguish between the first and the second.) They are: (12,13), (12,14), (12,15), (12,16), (12,17), (12,18), (13,12), (13,14), (13,15), etc.
What's the Difference?
When we sample with replacement, the two sample values are independent. Practically, this means that what we get on the first one doesn't affect what we get on the second. Mathematically, this means that the covariance between the two is zero.
In sampling without replacement, the two sample values aren't independent. Practically, this means that what we got on the for the first one affects what we can get for the second one. Mathematically, this means that the covariance between the two isn't zero. That complicates the computations. In particular, if we have a SRS (simple random sample) without replacement, from a population with variance , then the covariance of two of the different sample values is , where N is the population size. (A brief summary of some formulas is provided here. For a discussion of this in a textbook for a course at the level of M378K, see the chapter on Survey Sampling in Mathematical Statistics and Data Analysis by John A. Rice, published by Wadsworth & Brooks/Cole Publishers. There is an outline of an slick, simple, interesting, but indirect, proof in the problems at the end of the chapter.)
Replacement probability: We remove the element every time you choose one.
Without replacement: You don't replace the first item choose before you choose a second.
Step-by-step explanation:
Consider the provided information.
Replacement probability: In the probability of replacement we want to consider the probability of some case where there are a number of balls, cards or other items and you remove the element every time you choose one.
Without replacement: You don't substitute the first thing you choose in the without replacement before you choose a second.
#Learn more
A box containing 5 green balls and 3 red colour balls. Find the probability of selecting 3 green colour balls one by one (i) without replacement (ii) with replacement.
https://brainly.in/question/11626309